What is the effect on the Berry phase?

[jeon3133]

Homework Statement

Consider a Hamiltonian H[s] that depends on a number of slowly varying parameters collectively called s(t). What is the effect on the Berry phase γn[C] for a given closed curve C, if H[s] is replaced with f[s] H[s], where f[s] is an arbitrary real numerical function of the s?

Relevant Equations

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Homework Statement :
Consider a Hamiltonian H[s ] that depends on a number of slowly varying parameters collectively called s(t). What is the effect on the Berry phase γ_n[C] for a given closed curve C, if H[s ] is replaced with f[s ] H[s ], where f[s ] is an arbitrary real numerical function of the s?


Homework Equations :
For any s, we can find a complete orthonormal set of eigenstates Φ_n of H with eigenvalues E_n(s):
HΦ_n = E_nΦ_n
(Φ_n, Φ_m) = δ_nm.


Attempt at a Solution :
Could you help me to solve this problem?
Please...

 

[Dr_Nate]

Do you know the equation for the Berry phase?

 

[jeon3133]

In the special case where i and j run over three values,
γ_n[C] = ∫∫_A[C] dA e[s ] ⋅ V_n[s ], ----- (1)
where e[s ] is the unit vector normal to the surface A[C] at the point s, and V_n[s ] is a three-vector in parameter space:
V_n[s ] ≡ i ∑_m≠n{(Φ_n[s ], [∇H [s ]] Φ_m[s ])^* × (Φ_n[s ], [∇H [s ]] Φ_m[s ])} × (E_m[s ] - E_n[s ])^-2.

 

[Dr_Nate]

I don't understand. Is the closed curve given or is it arbitrary? In the Aharonov-Bohm effect, do you not get different answers if your integration encloses or doesn't enclose the solenoid?

Why do you think that you are told the function is slowly varying?